3.30.74 \(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4} (d+c x^4)} \, dx\)

Optimal. Leaf size=376 \[ -\sqrt {2} a^{3/2} \text {RootSum}\left [\text {$\#$1}^8 c-16 \text {$\#$1}^6 a^2 d+4 \text {$\#$1}^6 b c+32 \text {$\#$1}^4 a^2 b d+6 \text {$\#$1}^4 b^2 c-16 \text {$\#$1}^2 a^2 b^2 d+4 \text {$\#$1}^2 b^3 c+b^4 c\& ,\frac {\text {$\#$1}^4 \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )-2 \text {$\#$1}^2 b \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )+b^2 \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{\text {$\#$1}^6 c-12 \text {$\#$1}^4 a^2 d+3 \text {$\#$1}^4 b c+16 \text {$\#$1}^2 a^2 b d+3 \text {$\#$1}^2 b^2 c-4 a^2 b^2 d+b^3 c}\& \right ] \]

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Rubi [F]  time = 4.73, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)),x]

[Out]

Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d^(1/4) - Sqrt[-Sqrt[-c]]*x)*Sqrt[b + a^2*x^4]), x]/(4*d^(3/4)) +
Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d^(1/4) + Sqrt[-Sqrt[-c]]*x)*Sqrt[b + a^2*x^4]), x]/(4*d^(3/4))
+ Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d^(1/4) - (-c)^(1/4)*x)*Sqrt[b + a^2*x^4]), x]/(4*d^(3/4)) + De
fer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d^(1/4) + (-c)^(1/4)*x)*Sqrt[b + a^2*x^4]), x]/(4*d^(3/4))

Rubi steps

\begin {align*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, dx &=\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {d}-\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}} \, dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt {d}+\sqrt {-c} x^2\right ) \sqrt {b+a^2 x^4}} \, dx}{2 \sqrt {d}}\\ &=\frac {\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}-\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}+\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}}\right ) \, dx}{2 \sqrt {d}}+\frac {\int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}-\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{d} \left (\sqrt [4]{d}+\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}}\right ) \, dx}{2 \sqrt {d}}\\ &=\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}-\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}+\sqrt {-\sqrt {-c}} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}-\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (\sqrt [4]{d}+\sqrt [4]{-c} x\right ) \sqrt {b+a^2 x^4}} \, dx}{4 d^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4} \left (d+c x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)),x]

[Out]

Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)), x]

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IntegrateAlgebraic [A]  time = 12.09, size = 376, normalized size = 1.00 \begin {gather*} -\sqrt {2} a^{3/2} \text {RootSum}\left [b^4 c+4 b^3 c \text {$\#$1}^2-16 a^2 b^2 d \text {$\#$1}^2+6 b^2 c \text {$\#$1}^4+32 a^2 b d \text {$\#$1}^4+4 b c \text {$\#$1}^6-16 a^2 d \text {$\#$1}^6+c \text {$\#$1}^8\&,\frac {b^2 \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-2 b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text {$\#$1}^2+16 a^2 b d \text {$\#$1}^2+3 b c \text {$\#$1}^4-12 a^2 d \text {$\#$1}^4+c \text {$\#$1}^6}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)),x]

[Out]

-(Sqrt[2]*a^(3/2)*RootSum[b^4*c + 4*b^3*c*#1^2 - 16*a^2*b^2*d*#1^2 + 6*b^2*c*#1^4 + 32*a^2*b*d*#1^4 + 4*b*c*#1
^6 - 16*a^2*d*#1^6 + c*#1^8 & , (b^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt
[b + a^2*x^4]] - #1] - 2*b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x
^4]] - #1]*#1^2 + Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1
]*#1^4)/(b^3*c - 4*a^2*b^2*d + 3*b^2*c*#1^2 + 16*a^2*b*d*#1^2 + 3*b*c*#1^4 - 12*a^2*d*#1^4 + c*#1^6) & ])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2)/(c*x^4+d),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{4} + d\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2)/(c*x^4+d),x, algorithm="giac")

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/(sqrt(a^2*x^4 + b)*(c*x^4 + d)), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}\, \left (c \,x^{4}+d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2)/(c*x^4+d),x)

[Out]

int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2)/(c*x^4+d),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{4} + d\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2)/(c*x^4+d),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/(sqrt(a^2*x^4 + b)*(c*x^4 + d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\left (c\,x^4+d\right )\,\sqrt {a^2\,x^4+b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/((d + c*x^4)*(b + a^2*x^4)^(1/2)),x)

[Out]

int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/((d + c*x^4)*(b + a^2*x^4)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} \left (c x^{4} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2+(a**2*x**4+b)**(1/2))**(1/2)/(a**2*x**4+b)**(1/2)/(c*x**4+d),x)

[Out]

Integral(sqrt(a*x**2 + sqrt(a**2*x**4 + b))/(sqrt(a**2*x**4 + b)*(c*x**4 + d)), x)

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